Find the Asymptotes (-4x)/(x^2+4)

Solution:

Step 1:

Identify the values for which $\frac{-4x}{x^2 + 4}$ is not defined. The function is defined for all real numbers since the denominator never equals zero.

Step 2:

Determine if there are any vertical asymptotes by finding points of infinite discontinuity. There are no vertical asymptotes for this function.

Step 3:

Examine the rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

• If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.

• If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.

• If $n > m$, there are no horizontal asymptotes, but there may be an oblique asymptote.

Step 4:

Calculate the degrees $n$ and $m$.

• $n = 1$
• $m = 2$

Step 5:

Since $n < m$, the horizontal asymptote is the x-axis, which is $y = 0$.

Step 6:

There is no oblique asymptote since the degree of the numerator is less than the degree of the denominator.

Step 7:

Summarize the asymptotes for the function:

• No Vertical Asymptotes
• Horizontal Asymptote: $y = 0$
• No Oblique Asymptotes

Step 8:

Knowledge Notes:

To find the asymptotes of a rational function, one must understand the relationship between the degrees of the numerator and denominator. Here are the relevant knowledge points:

1. Domain of a Function: The set of all possible input values (x-values) for which the function is defined. A function is undefined where the denominator is zero.

2. Vertical Asymptotes: These occur at values of x where the function approaches infinity. They are found by setting the denominator equal to zero and solving for x, but only if these points are not canceled out by the numerator.

3. Horizontal Asymptotes: These are horizontal lines that the graph of a function approaches as x goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator:

   • If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), the horizontal asymptote is $y = 0$.

   • If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.

   • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

4. Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. They can be found by performing polynomial long division.

5. Degrees of Polynomials: The degree of a polynomial is the highest power of the variable in the polynomial. It is important to compare the degrees of the numerator and the denominator to determine the behavior of the rational function at infinity.

In the given problem, since the degree of the numerator ($n = 1$) is less than the degree of the denominator ($m = 2$), the horizontal asymptote is the x-axis, or $y = 0$. There are no vertical or oblique asymptotes because the denominator does not have real roots and the degree of the numerator is not greater than the degree of the denominator.